Convergence to pushed fronts and the behavior of level sets in monostable reaction-diffusion equations

We study the behavior of solutions of a monostable reaction-diffusion equation $u_t=Δ_x u +u_{yy} +f(u)$ ($x \in \mathbb{R}^{n-1}$, $y \in \mathbb{R}$, $t>0$), with the unstable equilibrium point $0$ and the stable equilibrium point $1$. Under the condition that the corresponding one-dimensional equation has a pushed front $Φ_{c^}(z)$ with $Φ_{c^}(-\infty)=1$, $Φ_{c^}(\infty)=0$, we show that the solution $u(x,y,t)$ approaches $Φ_{c^}(y-γ(x,t))$ for some $γ(x,t)$ as $t \to \infty$, if initially $u(x,y,0)$ decays sufficiently fast as $y \to \infty$ and is bounded below by some positive constant near $y=-\infty$. It is also shown that $γ(x,t)$ is approximated by the mean curvature flow with a drift term.

💡 Research Summary

The paper investigates the long‑time behavior of solutions to the n‑dimensional monostable reaction‑diffusion equation

  u_t = Δ_x u + u_{yy} + f(u),  x∈ℝ^{n‑1}, y∈ℝ, t>0,

under the assumption that the associated one‑dimensional equation possesses a pushed travelling front Φ_{c*}(z) with Φ_{c*}(−∞)=1 and Φ_{c*}(∞)=0. The reaction term f satisfies the standard monostable conditions (F): f(0)=f(1)=0, f′(0)>0, f′(1)<0, f(s)>0 for s∈(0,1), and f(s)<0 outside